As you've probably seen, the 1880 census has their two eldest children born in Ireland:
"United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MWFL-7FB : 15 July 2017), Patrick Carden, Carbondale, Lackawanna, Pennsylvania, United States; citing enumeration district ED 24, sheet 25B, NARA microfilm publication T9 (Washington, D.C.: National Archives and Records Administration, n.d.), FHL microfilm 1,255,137
Civil registration of births, marriages and deaths started in Ireland in 1864 so James should be listed and I was able to locate his birth record:
"Births registered in the district of Easky in the union of Dromore West in the county of Sligo".
"19th January 1864, Calhagh, James, Male, Patt Carden of Calhagh, Eliza Carden formerly Scott, Farmer, The mark of Molly Scott present at birth, Ballycummin, 28 January 1864, and the Registrar's signature"
You can find Easky and Ballycummin just three or four miles from Dromore West but I'm not clear about "Calhagh".